ravel wrote:I understand your way of looking at ALS, where you only need one set to make the conclusion. Once the set is found, it is an elegant way for showing, why an elimination can be made safely. Do you have an example, where you can make an elimination, which cannot be done with an (xz- or xy-) ALS also ?

I am afraid I don't know yet what xz- and xy-ALS are. I just noticed that each time people here invoke ALS, I would have used a simpler subset counting.

ravel wrote:And are you finding the set "as a whole" or are you putting it together from (2 or more) almost locked sets?

As a whole.

ravel wrote:So please can you show an x-wing elimination with subset counting and tell me, if a comprehensive forcing chain can be expressed with subset counting. For the latter i took an arbitrary sample:

- Code: Select all
` 3 5678 57 | 4 19* 26 | 27* 289 157#`

17 567 4 | 8 19* 26 | 3 29* 157

2 18 9 | 3 5 7 | 6 18 4

r1c9=7 => r1c7=2 => r2c8=9 => r2c5=1 => r1c5=9 => r1c9=1

Earlier today I answered flip on a similar question. This really belongs in a different thread - such things can be done by counting arguments, but not by the type of subset counting shown in the example. The chain is beautifully short, the counting reasoning is longer since both lower and upper bounds are involved. Roughly: assuming (1,9)7 one finds for the entire subset 129 with max multiplicities 112, and for row 1 min multiplicities 110, leaving max multiplicities 002 for row 2, impossible.